734 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
17. After introducing a new term containing (sg—rff), the equation of the complex 
may be written thus. 
Ar+Bs+C-)-D(7+Eg>+F(5§— r<r) = 0. 
When, after (ra— sg) is eliminated by means of the equation 
( 7 ) 
ry'—sx , =r<r—s$. 
we proceed as we did in the former case [14], the following equation is obtained in order 
to represent the plane corresponding to the given point ( od , y\ z'), 
(A-Fy-E^X*-^)+(B+F^-D^Xy-y)+(C+E^+I^X*-^)=0. . (8) 
This equation may be expanded thus, 
(A-Fy-E^>+(B+r^-D2 , >+(C+Ea/+Dy>=A^+By+C/, . . (9) 
and reduced also to the following symmetrical form, 
A(x — x ') + B (y — y ) -f C(z — z ') + D (y'z — z'y ) + E (a/z — z'x ) + F(x ! y —y'x) = 0 . (10) 
18. We may directly prove that all rays confined within a given plane meet in the 
same point. The equation of this plane being 
t'x-\-u'y-{-v'z-\- , w'= 0 , . ( 11 ) 
we get, in order to express that a ray falls within that plane, the following three equa- 
tions, 
i}r-\-v!s-\-v' =0, 
1j Q ?{/ = 0, 
w's—v'a—{rG—sq)t'= 0, 
each of which results from the other two. Between these equations and the equation 
of the complex (r<r—sg), r and % may be eliminated. The resulting equation, 
(B^-A«i'-Fw , )s + (Dif-Ew , + Fi;> + C^-Aw'-Ew'=0, . . . (12) 
being linear with regard to the two remaining variables s and <r, represents a right line 
parallel to OX and intersecting YZ in a point, the coordinates of which are 
, Bt’-Au'—Fw 1 ' 
Df*— Ett'— Ft/ ’ n o\ 
, Ct'-Av'-Bw' 
y—Ds-Ku'+w ' 
Hence all rays of the complex supposed to fall within the plane (11) intersect that right 
line, and consequently meet in the same point. Two coordinates of that point are given 
by the last equations, the third, 
, Cu'—Bv'—DuJ } 
X Dt* — Em' + Ft/ ’ / 
( 14 ) 
is obtained by introducing the values of z' and y 1 into the equation of the plane. 
